Hamiltonicity of Bell and Stirling Colour Graphs

For a graph $G$ and a positive integer $k$, the $k$-Bell colour graph of $G$ is the graph whose vertices are the partitions of $V$ into at most $k$ independent sets, with two of these being adjacent if there exists a vertex $x$ such that the partitions are identical when restricted to $V - \{x\}$. The $k$-Stirling Colour graph of $G$ is defined similarly, but for partitions into exactly $k$ independent sets. We show that every graph on $n$ vertices, except $K_n$ and $K_n - e$, has a Hamiltonian $n$-Bell colour graph, and this result is best possible. It is also shown that, for $k \geq 4$, the $k$-Stirling colour graph of a tree with at least $k+1$ vertices is Hamiltonian, and the 3-Bell colour graph of a tree with at least 3 vertices is Hamiltonian.